
Then, RCG (Q) updates \(\bar{{\cal X}}_t\)  along the conjugate direction, followed by retraction.
Regarding the step size, the optimal choice of \(\alpha _t\)  would be the minimizer of the objective function: \(\alpha _t = \arg \min _{\alpha \in \mathbb {R}}\bar{f}\left(\bar{R}^Q_{\bar{{\cal X}_t}}\left(\alpha \bar{\eta }_t\right)\right)\) .
However, for the retraction (REF ), the exact \(\alpha _t\)  is expensive to calculate, since the minimization problem is a degree \(d^2\)  polynomial in \(\alpha \) . Inspired by [1]}, we instead consider a degree 2 polynomial approximation of the minimization problem,
\(\arg \min _{\alpha \in \mathbb {R}}~\left\Vert  {\cal P}_{\Omega }\left(\phi \left(\bar{{\cal X}}_t\right)+\alpha \sum _{k = 1}^d\phi \left(\left\lbrace {\cal X}_t^1,\cdots ,\eta _t^k,\cdots ,{\cal X}_t^d\right\rbrace \right)\right)-{\cal P}_{\Omega }\left({\cal T}\right)\right\Vert _{{\footnotesize {\mathsf {F}}}}^2,\) 
